The Cost of EFX: Generalized-Mean Welfare and Complexity Dichotomies with Few Surplus Items
Eugene Lim, Tzeh Yuan Neoh, Nicholas Teh
TL;DR
This paper investigates the cost of enforcing Envy-Freeness up to any good (EFX) in fair division with few surplus items, where $m=n+c$ and $c\le 3$, a regime where existence of EFX is guaranteed. It unifies welfare analysis via generalized-mean welfare $W_p$ and reveals a sharp complexity split at $p=0$: for $p\in(0,1]$, maximizing or deciding global $W_p$-optimality under EFX is NP-hard even with $c\le 3$, while for $p\le 0$ there are polynomial-time algorithms to optimize within the EFX set and to certify global optimum. The work also quantifies the price of EFX, showing linear welfare loss in $n$ for $p>0$ but only constant-factor loss for $p\le 0$ (with Nash welfare approaching neutrality as $n$ grows). Additionally, enforcing Pareto-optimality with EFX remains hard ($\text{NP}$-hard; $\Sigma_2^P$-complete for a stronger variant), and the authors extend results under nonzero marginal utilities (NMU) and parameterized surplus $c$, yielding tractability in several structured settings. Overall, the paper delineates when EFX aligns with welfare maximization in the few-surplus-items regime and when it becomes computationally costly, informing algorithm design and fairness guarantees in practice.
Abstract
Envy-freeness up to any good (EFX) is a central fairness notion for allocating indivisible goods, yet its existence is unresolved in general. In the setting with few surplus items, where the number of goods exceeds the number of agents by a small constant (at most three), EFX allocations are guaranteed to exist, shifting the focus from existence to efficiency and computation. We study how EFX interacts with generalized-mean ($p$-mean) welfare, which subsumes commonly-studied utilitarian ($p=1$), Nash ($p=0$), and egalitarian ($p \rightarrow -\infty$) objectives. We establish sharp complexity dichotomies at $p=0$: for any fixed $p \in (0,1]$, both deciding whether EFX can attain the global $p$-mean optimum and computing an EFX allocation maximizing $p$-mean welfare are NP-hard, even with at most three surplus goods; in contrast, for any fixed $p \leq 0$, we give polynomial-time algorithms that optimize $p$-mean welfare within the space of EFX allocations and efficiently certify when EFX attains the global optimum. We further quantify the welfare loss of enforcing EFX via the price of fairness framework, showing that for $p > 0$, the loss can grow linearly with the number of agents, whereas for $p \leq 0$, it is bounded by a constant depending on the surplus (and for Nash welfare it vanishes asymptotically). Finally we show that requiring Pareto-optimality alongside EFX is NP-hard (and becomes $Σ_2^P$-complete for a stronger variant of EFX). Overall, our results delineate when EFX is computationally costly versus structurally aligned with welfare maximization in the setting with few surplus items.